The supremum in the middle is not essential -- with a little effort you can rearrange that inequality so that there are suprema is on the outside and one Euclidean norm in the middle. Really, for the purposes of solving for a and b, you probably just want to look at it as two separate inequalities.

Just draw a right triangle. Make the lengths of the legs x1 and x2. sqrt(x1^2+x2^2)=h is the length of the hypotenuse. In terms of h, what's the longest the longest leg can be? What's the shortest the longest leg can be?